Assign 7

course Mth 158

9/25 4:15

If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution:

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

007. `* 7

* R.7.10 (was R.7.6). Show how you reduced (x^2 + 4 x + 4) / (x^4 - 16) to lowest terms.

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Your solution:

x^2 + 4x + 4 First I reduced the numerator to

(x +2) (x + 2)

x^2 + 2^2

x^4 – 16 Then the denominator is reduced to

(x^2 – 4) (x^2 + 4)

(x-2) (x+2) (x^2 + 4)

So I re-write it to

(x+2)(x+2) / (x-2)(x+2)(x^2+4) and see that x+2 cancels each other leaving us with

x+2 / (x-2) (x^2+4)

confidence rating: 3

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Given Solution:

* * ** We factor the denominator to get first (x^2-4)(x^2+4), then (x-2)(x+2)(x^2+4). The numerator factors as (x+2)^2. So the fraction is

• (x+2)(x+2)/[(x-2)(x+2)(x^2+4)],

which reduces to

• (x+2)/[(x-2)(x^2+4)].

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Self-critique (if necessary):

I know it takes me the long way around to get the answer, but for now it’s the only way I can do it. I keep reading the chapters over and over and each time I understand a little more. Hopefully, it will just come to me and I will be able to work the problem as easily as the given solution.

you appear to be doing OK

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Self-critique Rating: 3

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Question:

* R.7.28 (was R.7.24). Show how you simplified[ ( x - 2) / (4x) ] / [ (x^2 - 4 x + 4) / (12 x) ].

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Your solution:

[ (x-2) / (4x) ] / [ (x^2 – 4x + 4) / (12x) ]

This is hard for me to explain since I have to write it on paper to even understand how it is laid out.

I solve it by taking

(x-2) / (4x) multiplying by the inverse of the denominator (12x) / (x^2 – 4 + 4)

Then I bring the two together as

(x-2) (12x) / (4x) (x^2 -4 +4) reducing the (x^2 -4 +4) to (x + 2^2)

Leaving me with

(12x) / (4x) ( x +2^2) = 3 / x -2

confidence rating: 2

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Given Solution:

[ ( x - 2) / (4x) ] / [ (x^2 - 4 x + 4) / (12 x) ] =

(x-2) * / 4x * 12 x / (x^2 - 4x + 4) =

(x-2) * 12 x / [ 4x ( x^2 - 4x + 4) ] =

12 x (x-2) / [4x ( x-2) ( x-2) ] =

3/(x - 2) **

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Self-critique (if necessary):

There are no discrepancies.

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Self-critique Rating: 3

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Question:

* R.7.40 (was R.7.36). Show how you found and simplified the sum (2x - 5) / (3x + 2) + ( x + 4) / (3x + 2).

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Your solution:

(2x – 5) / (3x + 2) + (x + 4) / (3x + 2) Since we have the same value as both denominators it would be written

(2x – 5) + (x + 4) / (3x + 2) = (3x – 1) / (3x + 2)

confidence rating: 3

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Given Solution:

We have two like terms so we write

• (2x-5)/(3x+2) + (x+4)/(3x+2) = [(2x-5)+(x+4)]/(3x+2).

Simplifying the numerator we have

• (3x-1)/(3x+2).

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Self-critique (if necessary):

There are no discrepancies.

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Self-critique Rating: 3

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Question:

* R.7.52 (was R.7.48). Show how you found and simplified the expression (x - 1) / x^3 + x / (x^2 + 1).

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Your solution:

(x-1) / x^3 + x / (x^2 + 1)

In this case you would multiply the first set by the x^2 + 1 / x^2 + 1

I don’t know how to set this up with typewriter notation, but I’ll try

(x-1) * (x^2 + 1) / (x^3) * (x^2 + 1) and then add that to (x) * (x^3) / (x^2 + 1) * (x^3)

Reducing it to

(x-1)(x^2 +1) / (x^3)(x^2 +1) + x^4 / (x^3)(x^2 +1)

x^3 + x – x^2 – 1 + x^4 / (x^3)(x^2 + 1)

As written your solution x^3 + x – x^2 – 1 + x^4 / (x^3)(x^2 + 1) indicates that you first divide x^4 by x^3, then multiply the result by x^2 + 1. You finally add this result to x^3 + x - x^2 - 1. I do not believe this is what you intended.

I believe every step in your solution was correct as you wrote it on paper, and that what you intended to say was correct.

However to put it correctly into typewriter notation, you need to group the denominator throughout, and in the last step you need to group your numerator.

Using brackets to emphasize the grouping of numerator and denominator:

[ x^3 + x – x^2 – 1 + x^4 ] / [(x^3)(x^2 + 1)]

We would then put the numerator into order of decreasing exponents, so arrive at an easily-recognized standard form:

[ x^4 + x^3 - x^2 + x - 1 ] / [(x^3)(x^2 + 1)]

confidence rating: 1

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Given Solution:

Starting with (x-1)/x^3 + x/(x^2+1) we multiply the first term by (x^2 + 1) / (x^2 + 1) and the second by x^3 / x^3 to get a common denominator:

[(x-1)/(x^3) * (x^2+1)/(x^2+1)]+[(x)/(x^2+1) * (x^3)/(x^3)], which simplifies to

(x-1)(x^2+1)/[ (x^3)(x^2+1)] + x^4/ [(x^3)(x^2+1)].

Since the denominator is common to both we combine numerators:

(x^3+x-x^2-1+x^4) / [ (x^3)(x^2+1)] .

We finally simplify to get

(x^4 +x^3 - x^2+x-1) / / [ (x^3)(x^2+1)]

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Self-critique (if necessary):

I didn’t simplify the last step.

I worked this on paper last night and then copied it over to word so I could post it, but when I was re-typing it, I couldn’t remember how I got the x^3 + x – x^2 – 1 + x^4. I am sure it is a formula, but I don’t remember which one I used. Can you tell me how to recognize what formula I need to use to solve a problem? I think that is where I’m spending most of my time trying to figure out what to do and looking it up in the text book.

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Self-critique Rating: 3

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Question:

* R.7.58 (was R.7.54). How did you find the LCM of x - 3, x^3 + 3x and x^3 - 9x, and what is your result?

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Your solution:

To find the LCM, you find everything that factors into each one of the problems. I look at it like a universal set and then the rest are subsets of that set.

x – 3 is just x – 3

x^3 + 3x factors to x ( x^2 + 3)

x^3 – 9x factors to x (x-3)(x+3)

Then I find each sub set and put them together to make a universal set

LCM = x (x-3) (x^2 +3) (x + 3)

confidence rating: 2

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Given Solution:

x-3, x^3+3x and x^3-9x factor into

x-3, x(x^2+3) and x(x^2-9) then into

(x-3) , x(x^2+3) , x(x-3)(x+3).

The factors x-3, x, x^2 + 3 and x + 3 'cover' all the factors of the three polynomials, and all are needed to do so. The LCM is therefore:

• LCM = x(x-3)(x+3)(x^2+3)

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Self-critique (if necessary):

I don’t know if my answer is acceptable in the way that I find the answer, but it is the only way I can understand how to solve it at the moment.

What you are doing is fine.

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Self-critique Rating: 1

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Question:

* R.7.64 (was R.7.60). Show how you found and simplified the difference 3x / (x-1) - (x - 4) / (x^2 - 2x + 1).

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Your solution:

3x / (x-1) – (x-4) / (x^2 – 2x + 1)

I got stuck on the first part of this one.

confidence rating: 0

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Given Solution:

* * ** Starting with

3x / (x-1) - (x-4) / (x^2 - 2x +1)

we factor the denominator of the second term to obtain (x - 1)^2. Since the first denominator (x - 1) is already a factor of the second, our common denominator is (x - 1)^2.

To express the given expression in terms of the common denominator we then multiply the first expression by (x-1) / (x-1) to get

3x(x-1)/(x-1)^2 - (x-4)/(x-1)^2,

which gives us

(3x^2-3x-x-4) / (x-1)^2 = (3x^2 - 4x - 4) / (x-1)^2.

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Self-critique (if necessary):

I don’t see where the 3x(x-1) comes in. When I tried working it on paper, I started with

3x / (x -1) – (x-4) / (x^2 - 2x + 1) and reduced to

3x / (x-1) – (x-4) / (x-1)^2

But then I tried to solve it by

(3x) – (x-4) / x-1 which isn’t right

3x / (x-1) – (x-4) / (x-1)^2 consists of two fractions with different denominators. You can't do the subtraction until you have a common denominator.

The denominator of the first is (x - 1); the denominator of the second is (x - 1)^2. If you multiply the first denominator by (x - 1) you get the second denominator.

So to express the first fraction with the same denominator as the second you multiply its numerator and denominator both by ( x - 1 ). You get

(3x / (x - 1) ) * (x - 1) / (x - 1) = 3x ( x - 1) / (x - 1)^2.

So the subtraction becomes

3x ( x - 1) / ( x - 1) ^ 2 - (x - 4) / (x - 1)^2.

The denominator of the combined fraction is (x - 1)^2, while the numerator is (3 x * (x - 1) - (x - 4) ); this simplifies to 3 x^2 - 3 x - x + 4, whic gives us

(3x^2-3x-x-4) / (x-1)^2 = (3x^2 - 4x - 4) / (x-1)^2.

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Self-critique Rating:

QUESTION FROM STUDENT: On the practice test I'm having problems with problem #5 I don't know where to start or how to set it up. I'm probably missing something simple and will probably feel stupid by seeing the solution. Could you help with this problem.

A retailer is offering 35% off the purchase price of any pair of shoes during its annual charity sale. The sale price of the shoes pictured in the advertisement is $44.85. Find the original price of the shoes by solving the equation p-.35p = 44.85 for p.

INSTRUCTOR RESPONSE: It's very easy to grab onto the wrong idea on a problem and then have trouble shaking it, or to just fail to look at it the right way. Nothing stupid about it, just human nature.

See if the following makes sense. If not let me know.

p - .35 p = 44.85. Since p - .35 p = 1 p - .35 p = (1 - .35) p = .65 p we have

.65 p = 44.85. Multiplying both sides by 1/.65 we get

p = 44.85 / .65 = etc. (you can do the division on your calculator); you'll get something near $67).

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&#This looks good. See my notes. Let me know if you have any questions. &#